1. Field of the Invention
The present invention relates to the art of design load bearing design members and, more particularly, to the method of using data of deformation and geometrical stiffness for calculating the elastic limit and corresponding optimal dimensions of a member.
The present invention also relates to nondestructive testing resistance of a member to elastic deformation and, more particularly, to the testing geometrical stiffness of a member.
The invention also relates to the testing mechanical properties of the structures and materials.
2. Description of the Prior Art
The prior art of design is based on well known theories of strength such as maximum-stress theory, maximum-strain theory, maximum strain-energy theory. A physical concept underlying these theories is that material limits the application of Hooke's Law of elasticity, .sigma.=E.epsilon.(1). According to the most common maximum-stress theory member is considered to be reliable if maximum stress in the member is less than proportional limit of the material.
"Hooke's Law. It has been found by experiment that a body acted on by external forces will deform in proportion to the stress developed as long as the unit stress does not exceed a certain value which varies for the different materials. This value is the proportional limit." (Eshbach, Handbook of Engineering Fundamentals, 3d Ed.,p. 489) PA1 "Equation is known as Euler's column formula indicates that the critical buckling load is not a function of the strength of the material (yield and ultimate strengths are not envolved) but only of elastic modulus and geometry." (Engineering Design, by Faupel and Fisher, 2nd Ed.,p. 568) PA1 "Fundamental data obtained in a test on material are affected by the method of testing and the size and shape of the specimen. To eliminate variations in results due to these causes, standards have been adapted by ASTM, ASME and various associations and manufactures." (Eshbach, Handbook of Engineering Fundamentals, 3d ED.,p. 566) PA1 "This assumption is not true for very short columns, nor it is true for columns of medium length such as are usually needed in practice. There is no exact formula which gives the strength of a column of any length under an axial load." (Eshbach, Handbook of Engineering Fundamentals, 3d Ed.,p. 529) PA1 "Here the continuing trend towards lighter and thinner structures associated with the use of high strength material is bringing problems of elastic stability increasingly to the fore. This has long been the case in aerospace field, but it is now rapidly extending to ships and to high-rise buildings. And as designs become even more efficient the engineer will be faced with even more instabilities demanding the sophisticated treatments." (A General Theory of Elastic Stability, by J. M. T. Thompson and G. W. Hunt, 1971 London,p. 48).
Then, the art of calculating dimensions of a member follows the theory. Stresses in the member can be obtained analytically or by measurement. Test of material using the standard specimen gives mechanical properties of the material such as proportional limit, elastic limit, ultimate strength, modulus of elasticity of material. Maximum stress in the member then compared with proportional limit of the material for calculating the cross-sectional characteristics or correcting them. For example, in case of tension, .sigma.=N/A .ltoreq.[.sigma..rho.] (2), A.gtoreq.N/[.sigma..sub..rho. ] (3). In the case of bending, .sigma..sub.max =M.sub.max /Z.ltoreq.[.sigma..sub..rho. ](4), Z.gtoreq.M.sub.max /[.sigma..sub..rho. ](5)
Presently, there is no universality in the theories of strength. In contrary to the general strength theories the theory of buckling is based on assumption that critical buckling load or stress does not depend on the critical characteristics of the material, but depends on geometry and modulus of elasticity of material only.
The prior art of design has great achievements. However, it has major flaws as well. The main disadvantage of the prior art is that strength theories do not corroborate well with the physical evidence. Thus, the strength theories contradict to overwhelming evidence that critical for a structure load or stress depends on geometry of design and modulus elasticity of material and not a function of the material strength.
It is true for the buckling and all general cases of deformation as well. Confirmation to this point of view can be found in the tests of materials and structures.
It is obvious that eliminate differences in the size, shape, method of loading is impossible in the structures other than specimen. Different structures made of the same material have different limits. The standards and tests of the structures of some but insufficient help. Common physical foundation and the equations describing relations between critical for the design load and the geometry of the design must be developed.
The buckling empirical formulas developed for different practical cases are not applicable for general cases of bending, tension, torsion. In fact, these formulas are not very reliable even for cases of buckling. The buckling formulas are developed with the assumption that failure of the columns, for example, occurs due to the sidewise bending.
The theories of strength remained hypothetical for centuries. And design technic became more and more complicated due to uncertainty in the art of design.
Examplary teachings of prior art of design and assisting devices can be found in the textbooks, handbooks, American Standards, U.S. Pat. Nos. 4,788,868; 4,884,453; 4,884,445; 4,884,455, also in the article `Deflections Indicate Design Similarity` by L. Ratner, `Machine Design` of May 8, 1986.
The present invention in the art of design is based on a new and different physical concept of strength. According to this concept each structure has an individual proportional and elastic limits which, in general, are different from the limits of the material. The limit of a structure depends on the resistance of a structure to elastic deformation. Physical characteristic describing this resistance is called `stiffness`. Stiffness depends on elasticity of the material(E), geometry of the design and boundary conditions.
A part of the stiffness which is a function of the size, shape, specific design features and boundary condition is singled out and described as a new important characteristic of a structure called `geometrical stiffness`. Then, general equation of elastic deformation can be written as following. Total elastic deformation is proportional to the distributed in the structure force and inversely proportional to the geometrical stiffness and modulus elasticity of material, D=F/ER (6).
Thus, in case of tension, e=N/ER (7) where R=KA/L (8). In case of bending total angular deformation, .theta.=M/ER (9) where geometrical stiffness R=KI/L (10). Geometrical stiffness in the equations of elastic deformation is presented as a physical entity.
The equations of deformation in the prior art are different and the difference is not formal but of practical importance. Presently, in case of tension, e=NL/EA (11.). In case of bending the general equation of elastic line, M=.+-.EI*dY.sup.2 /dx.sup.2. From the general equation the equations for different specific cases are developed. For example, for the simple beam with concentrated load at the center, Y.sub.max =PL.sup.3 /48EI (12), .theta..sub.max =PL.sup.2 /16EI (13).
For the purpose of optimization of dimensions it is necessary to know how geometry, in particular size, affects deformation. The equation which should show such effect is differential equation derived from the equation of elastic deformation. Scientific logic suggests that if an equation represents the relations between components correctly then derivative differential equation will be also correct. However, it appears that differential equations derived from existing equations of deformation are incorrect. For example, d.theta./dI=-PL.sup.2 /16EI.sup.2, does not describe the rate of change of deformation depending on change of moment of inertia of cross-section correctly. The equations of elastic deformation in the prior art are unsuitable for the purpose of optimization. The new equations of deformation are different. The main components in the equation are distributed in a structure forces, geometrical stiffness, total deformation. Though, each of these components can be presented as a function in the equation of deformation said components presented as the physical entities. This holistic approach differs from existing disintegrated approach when the equation of deformation became the mixture of elements belonging to the components of different physical origin. Thus, in the equation Y.sub.max =PL.sup.3 /48EI one of L-s belongs to the bending moment, other to the geometrical characteristic, yet another to the resulting deformation. Such attitude of neglecting physical meaning of components lead to the flaws in representation of relations and in results. New equations describe elastic relations more accurately. Differential equations derived from them are also correct. On example of a beam deformation--geometrical stiffness relation is presented graphically in diagram .theta. vs. R (FIG. 1). The diagram shows rapid increase of deformation in the interval proportional-elastic limit. Beyond this limit, an insignificant decrease in stiffness results in failure of elastic behavior. Likewise, increasing geometrical stiffness above the proportional limit does not improve elastic stability. The point R.sub.a on diagram shows position of an actual geometrical stiffness of tested structure. The point R.sub.o shows position of optimal geometrical stiffness for given force and material.
The rate of change of deformation due to geometrical stiffness is a reliable criterion for design optimization. The proportional and elastic limits are characterized with the rate of change of deformation. In the interval proportional- elastic limit the rate can be anticipated from tan .alpha.=1.0 (.alpha.=45.degree.) to tan .alpha.=3.7 (.alpha.=75.degree.).
The rate of change of deformation can be described with differential equation derived from the equation of elastic deformation, dD/dR=-F/ER.sup.2 (14). The relations within limits is F/ER.sup.2 =C.sub.s (15) where C.sub.s =tan .alpha. is coefficient of elastic stability. The equation (15) is a foundation of elastic design. Thus, in case of bending, d.theta./dR=M/ER.sub.o.sup.2 =C.sub.s (16). Optimal geometrical stiffness R.sub.o =(M/EC.sub.s).sup.0.5 (17).
Geometrical stiffness of a beam is a function of moment of inertia of cross-section, length, specifics of a beam design and boundary conditions, R=KI/L (10) where `K` is a coefficient which counts effect of specifics and boundary conditions on geometrical stiffness.
Physical meaning of geometrical stiffness is clear from the equation. The greater is moment of inertia, the greater is geometrical stiffness. The greater is length, the less is geometrical stiffness. Different beams may have the same stiffness if they have the same ratio moment of inertia to the length, R=KI.sub.1 /L.sub.1 =KI.sub.2 /L.sub.2.
Absolutely different structures may have the same geometrical stiffness, R=M/E.theta.. Considering geometrical stiffness as an entity, as a new property of a structure allows to establish standards of geometrical stiffness for the purpose of measurement. Note, that mode of deformation is considered in defining stiffness for bending, tension, torsion.
In order to correct geometrical stiffness of a beam, for example, one can change moment of inertia of cross-section. Here, R.sub.o /R.sub.a =I.sub.o /I.sub.a. Optimal moment of inertia, I.sub.o =I.sub.a R.sub.o /R.sub.a (18). The problem of calculating optimal moment of inertia with equation (10) is in the fact that coefficient `K` in the equation which counts specifics of design and boundary conditions initially can be obtained only experimentally. The series of similar structures have common coefficient `K`.
In some cases limit of elasticity of material can present limitation for a structure. Therefore, maximum stress in the structure of optimal dimensions must be checked against stress allowable by the material .sigma..sub.max =M.sub.max /Z.sub.o .ltoreq..vertline..sigma..sub..rho. .vertline. (19).
The method of optimization of load bearing design members which is further described on examplary teachings is an essential general method. The method can be used for optimization of any type of design of any complexity and of any material. The method is directed to optimization series of similar structures by testing one representative. It is very economical method. The new method will contribute to the safety and reliability of the structures and give savings on materials, labor, time.
The material presented makes it clear fundamental difference between the prior art of design and a new art and advantages of the new art. The prior art did not realize an existance the individual limit of a structure. It makes the methods of the prior art deficient. Further, in order to choose proper dimensions it is necessary to know how geometry affects behaviour of a structure. The rate of change of deformation is an indicator of elastic behavior. There is no knowledge of that in the prior art. Fixed criteria of limiting stress and limiting deformation in the prior art do not describe elastic behavior and they are unsuitable for the purpose of optimization.
There is no equation which describes rate of change of deformation depending on geometry in the prior art. Existing in the prior art equation of elastic: deformation cannot be used for that purpose for it does not describe relations deformation-geometry correctly.
The new property of a structure, e.g., geometrical stiffness is introduced in the art of design in order to reflect an effect of geometry on elastic behaviour. A new equation of elastic deformation describes deformation-force-geometrical stiffness relations. The derived equation describes rate of change of deformation depending on geometrical stiffness. Both equations essential for scientific design process are missing in the prior art.
An equation describing geometrical stiffness of a structure makes it possible to compare similar structures of different dimensions. It allows economical optimization series of similar structures after testing stiffness of a representative structure.
The new art challenges prior art. It makes it possible to compare structures, to predict behavior of structures, to make design process scientific rather than empirical.
Major differences between the prior art of design and new art are summarized in the TABLE OF COMPARATIVE ANALYSIS OF PRIOR ART AND THE NEW METHOD.
______________________________________ TABLE OF COMPARATIVE ANALYSIS OF PRIOR ART AND THE NEW METHOD Prior Art New Method ______________________________________ 1. Hooke's Law Strength of a structure is Each structure has an individual identified with strength of elastic limit which is different material, .sigma. = E .epsilon. = [.sigma..sub.p ]. from the limit of material and need to be considered in design process. 2. Equations of elastic deformation Tension: e = NL/EA e = N/ER where R = KA/L Bending: .theta. = PL/K'EI .theta. = M/ER where R = KI/L Torsion: .phi. = TL/GJ .phi. = T/GR where R = KJ/L In the equations of deformation `R` is a single characteristic called geometrical stiffness. 3. Equations of elastic stability The equations describing how N/ER.sub.o.sup.2 = C.sub.s ; M/ER.sub.o.sup.2 = C.sub.s ; T/GR.sub.o.sup.2 = geometry of a structure affects C.sub.s where `Cs` is a coefficient of deformation are not developed. elastic stability. Coefficient describes rate of change of deformation in the interval proportional-elastic limit. 4. Main criterion for design Certain characteristic of a Certain rate of change of deformation material such as proportional is a criterion of design. It is true limt is criterion of design. within elastic limit of material. .sigma. = N/A = [.sigma..sub.p ], A = N/[.sigma..sub.p ] R.sub.o = (N/EC.sub.s)0.5, R.sub.o = KA.sub.o /L .sigma. = M/Z = [.sigma..sub.p ], Z = M/[.sigma..sub.p ] R.sub.o = (M/EC.sub.s)0.5, R.sub.o = KI.sub.o /L. ______________________________________